Page 441 - Shigley's Mechanical Engineering Design

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416 Mechanical Engineering Design
Since we are not interested in the normal force N, we eliminate it from each of these
sets of equations and solve the result for P. For raising the load, this gives

F(sin λ + f cos λ)
P R = (c)
cos λ − f sin λ
and for lowering the load,

F( f cos λ − sin λ)
P L = (d)
cos λ + f sin λ
Next, divide the numerator and the denominator of these equations by cos λ and use
the relation tan λ = l/πd m (Fig. 8–6). We then have, respectively,
F[(l/πd m ) + f ]
P R = (e)
1 − ( fl/πd m )
F[ f − (l/πd m )]
P L = (f)
1 + ( fl/πd m )
Finally, noting that the torque is the product of the force P and the mean radius d m /2,
for raising the load we can write

Fd m l + π fd m
T R = (8–1)
2 πd m − fl
where T R is the torque required for two purposes: to overcome thread friction and to
raise the load.
The torque required to lower the load, from Eq. ( f ), is found to be

Fd m π fd m − l
T L = (8–2)
2 πd m + fl
This is the torque required to overcome a part of the friction in lowering the load. It may
turn out, in specific instances where the lead is large or the friction is low, that the load
will lower itself by causing the screw to spin without any external effort. In such cases,
the torque T L from Eq. (8–2) will be negative or zero. When a positive torque is
obtained from this equation, the screw is said to be self-locking. Thus the condition
for self-locking is

π fd m > l
Now divide both sides of this inequality by πd m . Recognizing that l/πd m = tan λ, we
get
f > tan λ (8–3)
This relation states that self-locking is obtained whenever the coefficient of thread
friction is equal to or greater than the tangent of the thread lead angle.
An expression for efficiency is also useful in the evaluation of power screws. If
we let f = 0 in Eq. (8–1), we obtain

Fl
T 0 = (g)
2π